Answer
The lengths of the parallelogram’s sides are $8.9\text{ inches}$ and $23.9\text{ inches}$.
Work Step by Step
Let $a$ be the side opposite to the angle $35{}^\circ $ and the other side be $c$.
The angle formed by a straight line is $180{}^\circ $. So, we will find out $\angle AOB$ as follows:
$\begin{align}
& \angle AOB=180{}^\circ -\angle BOC \\
& =180{}^\circ -35{}^\circ \\
& =145{}^\circ
\end{align}$
According to the cosine law,
${{a}^{2}}={{b}^{2}}+{{c}^{2}}-2bc\cos \theta $
Side $a$ of the triangle is
$\begin{align}
& {{a}^{2}}={{15}^{2}}+{{10}^{2}}-2\left( 15 \right)\left( 10 \right)\text{ }\cos 35{}^\circ \\
& {{a}^{2}}=79.3 \\
& a=\pm \sqrt{79.3} \\
& a=8.9
\end{align}$
Side c of the triangle is
$\begin{align}
& {{c}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\ \cos C \\
& {{c}^{2}}={{15}^{2}}+{{10}^{2}}-2\left( 15 \right)\left( 10 \right)\cos 145{}^\circ \\
& {{c}^{2}}=\pm \sqrt{570.7} \\
& c\simeq 23.9
\end{align}$
